Topological groups and C-embeddings

The notion of a Moscow space is applied to the study of some problems of topological algebra, following an approach introduced by A.V. Arhangelʹskii [Comment. Math. Univ. Carolin. 41 (2000) 585–595]. In particular, many new, and, it seems, unexpected, solutions to the equation νX×νY=ν(X×Y) are identified. We also find new large classes of topological groups G, for which the operations in G can be extended to the Dieudonné completion of the space G in such a way that G becomes a topological subgroup of the topological group μG. On the other hand, it was shown by A.V. Arhangelʹskii [Comment. Math. Univ. Carolin. 41 (2000) 585–595] that there exists an Abelian topological group G for which such an extension is impossible (this provided an answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985). Some new open questions are formulated.